Generating Bessel-Gaussian Beams with Controlled Axial Intensity Distribution

applied sciences

Article Generating Bessel-Gaussian Beams with Controlled Axial Intensity Distribution

Nikita Stsepuro 1 , Pavel Nosov 1 , Maxim Galkin 1,2, George Krasin 1 , Michael Kovalev 1,* and Sergey Kudryashov 2

1 Laser and Optoelectronic Systems Department, Bauman Moscow State Technical University, 2nd Baumanskaya st. 5/1, 105005 Moscow, Russia; [email protected] (N.S.); [email protected] (P.N.); [email protected] (M.G.); [email protected] (G.K.) 2 P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy Prospekt 53, 119991 Moscow, Russia; [email protected] * Correspondence: [email protected]

Received: 24 September 2020; Accepted: 5 November 2020; Published: 8 November 2020

Abstract: This paper investigated the diffraction of a Gaussian laser beam on a binary mask and a refractive axicon. The principles of the formation of a zero-order Bessel beam with sharp drops of the axial field intensity edges were discussed. A laser optical system based on an axicon for the formation of a Bessel beam with quasi-uniform distribution of axial field intensity was proposed. In the laser optical system, the influence of the axicon apex did not affect the output beam. The results of theoretical and experimental studies are presented. It is expected that the research results will have practical application in optical tweezers, imaging systems, as well as laser technologies using high-power radiation.

Keywords: Gaussian laser beam; Bessel beam; laser optical system; axicon; amplitude binary mask

1. Introduction Research on the methods of focusing light beams associated with the names of such scientists as Ernst Abbe, George Airy, and John Strutt have a long history and remain relevant to this day. Tight focusing of laser radiation, which allows one to obtain high power densities and directivity, is in demand in many problems of photonics and laser physics [1–6]. In practice, Gaussian laser beams are widely used. When they are focused using an optical system with spherical (or parabolic) surfaces, it is possible to concentrate laser radiation in the region corresponding to the beam waist diameter. However, in Gaussian beams, the wavefront is flat only in the waist region, the length of which is equal to the confocal parameter. With a decrease in the waist diameter of a Gaussian beam, its confocal parameter decreases [7–9]. By choosing the type and shape of the surfaces of the laser optical system elements, it is possible to correct the geometry of the wavefront. Correction methods can also be applied to increase the focusing length of the formed beam. In the end, when the lens takes the form of a cone, the phase front of the transmitted wave becomes conical. Such a conical lens is called an axicon [10]. The axicon forms a beam with a radial distribution of field intensity, which is described by the Bessel function of the first kind of the zero-order. Such beams are often used for scientific and applied problems [11,12], in which they have advantages over Gaussian beams [13,14]. The main ones are: (i) the transverse distribution of field intensity remains practically unchanged over a significant portion along the optical axis, (ii) the distribution of the field is restored behind obstacles, and (iii) excellent depth of field. Axicon is a reliable optical element for the formation of a zero-order Bessel beam, including schemes with high-power laser radiation. However, it has disadvantages such as the dependence of the

Appl. Sci. 2020, 10, 7911; doi:10.3390/app10217911 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7911 2 of 9 parameters of the generated beam on a specific element, the dependence of the intensity distribution of the output beam on the quality of element manufacturing (in particular, on the quality of the apex), and uneven distribution of axial intensity [15]. There is a huge number of different tools used for the formation of Bessel beams [16,17] which allow controlling axial intensity distribution. Diffraction [18,19] and holographic methods [20] are outside of the scope of this study due to the features of their implementation. Methods capable of realizing the required Bessel beam intensity profiles are widely used in microscopy [21], including biological imaging [22–24], as well as in applications of ultrashort laser pulses [25,26]. In almost all of the abovementioned applications, a Bessel beam with a controlled quasi-uniform axial intensity distribution is formed. The only way to change the axial intensity is to apodize the beam incident on the element (axicon or lens). In research, phase masks with a gradient, step, or both dependences of the phase shift value on the radius vector in combination with an axicon or a lens are actively used [27,28]. There are also sophisticated ways to achieve the same goal [29]. Research in the field of parametrization of the quasi-uniform axial intensity distribution of the Bessel beam is relevant. The purpose of this work was to develop a laser optical system for forming a Bessel beam with a controlled quasi-uniform axial distribution of field intensity on a standard element base with a minimum number of optical elements without significant restrictions on their radiation resistance. Research is being carried out on the possibility of realizing such a laser optical system by apodizing the input aperture of the axicon.

2. Modeling

The fundamental TEM00 mode formed by laser resonators of a stable conﬁguration corresponds to a Gaussian beam with a radially symmetric ﬁeld distribution [30]:

2 2 u (r, z) = A hw exp r exp i π r arctan z + kz , 00 w h(z) h2(z) λ R (z) zc q − − Φ − (1) 2 2 2 h(z) = hw 1 + (z/zc) , RΦ(z) = z 1 + zc /z . where u is the distribution of the complex amplitude of the Gaussian beam in the section with the z 00 p coordinate from the waist; r = x2 + y2 denotes the distance from the observation point in the beam to the z axis; x, y are the transverse coordinates; Aw denotes the amplitude constant or the field amplitude on the axis (r = 0) in the section of the beam waist (z = 0); h(z) and RΦ(z) are the dependences of the caustic and radius of wavefront curvature of the beam in section z; 2hw denotes the waist diameter of πh2 z = w λ M2 the Gaussian beam; c M2λ is Rayleigh length of the Gaussian beam; is laser wavelength; is the parameter quality of a Gaussian beam. Let us consider the transformation of a Gaussian beam with field distribution Equation (1) by a binary mask and a refractive axicon with a refractive index n and apex angle 2α0 (Figure1). In wave optics, the effect of an axicon is described by the complex amplitude transmittance [31]: τ(r) = τA(r)τΦ(r), where τA(r) is amplitude transmittance, τΦ(r) = exp(ikpr) is phase transmittance, p = sin β is the numerical aperture of the axicon, β = arcsin(n cos α ) + α π is the angle of 0 0 − 2 wavefront inclination to the optical axis after passing through the axicon in the case of the plane homogeneous wave. If the effect of the axicon has only a phase nature, then the generated Bessel beam of the zero-order is characterized by the length of the Bessel zone zB and the diameter of the core DB (diameter of the central maximum) of the field intensity distribution of the Bessel beam. They are defined by the following expressions [32,33]: h 2.405λ z = w , D = , (2) B tan β B 2π sin β Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 9

h 2.405λ Appl. Sci. 2020, 10, 7911 zD==w , 3 of 9 BBβ π β (2) tan 2 sin , Such a Bessel beam beam also also has has an an uneven uneven dist distributionribution of of axial axial field ﬁeld intensity intensity [33,34]: [33,34 ]:

8sinπβP zz2 22  ()8πP sin β z AP−2z AP  Iz0,AP ~AP expAP  (3) I(0, zAP) λ exp 2  (3) ∼ λh hzw z BB− z2  w B zB where zAP is the analysis plane after the axicon and P denotes the power of laser radiation. where zAP is the analysis plane after the axicon and P denotes the power of laser radiation.

Figure 1. Schemes for the formation of a zero-order Bessel beam with a quasi-rectangular dependence of axial intensity: (a) the Bessel zone starts right after the axicon; (b) the Bessel zone is moved away from the axicon. 2hw—waist diameter of the Gaussian beam, 2α0—apex angle axicon, n—refractive Figure 1. Schemes for the formation of a zero-order Bessel beam with a quasi-rectangular dependence index axicon, zAP—analysis plane after the axicon, zB—length of the Bessel zone, D—clear aperture of of axial intensity: (a) the Bessel zone starts right after the axicon; (b) the Bessel zone is moved away the binary mask, d—diameter of the inner aperture of the mask, τA—amplitude transmittance. α from the axicon. 2hw —waist diameter of the Gaussian beam, 2 0 —apex angle axicon, n —refractive

indexLet us axicon, investigate zAP —analysis the influence plane of after binary the maskaxicon, parameters zB —length and of the the Bessel input zone, Gaussian D—clear beam aperture on forming τ a quasi-rectangularof the binary mask, axial d—diameter intensity distribution of the inner aperture of the zero-order of the mask, Bessel A —amplitude beam. Amplitude transmittance. transmittance τA(r) is determined by the finite axicon aperture and the nature of its transmission within the clear aperture.Let us The investigate calculation the ofinfluence the transverse of binary distribution mask parameters of field intensityand the ininput the Gaussian analysis planebeamz onAP formingafter the a axicon quasi-rectangular is performed axial using intensity a certain distri transferbution operator. of the zero-order The choice Bessel of the beam. transfer Amplitude operator depends on theτ distance() between the original plane and the analysis plane. The nonparaxial scalar transmittance A r is determined by the finite axicon aperture and the nature of its transmission withinmodel the based clear on aperture. Rayleigh–Sommerfeld The calculation theoryof the transverse allows to distribution obtain correct of field results intensity at suffi inciently the analysis close distances from the aperture of the optical element at which the diffraction occurs. Using the integral plane zAP after the axicon is performed using a certain transfer operator. The choice of the transfer Rayleigh–Sommerfeld transform of the first type, we find the amplitude-phase distribution of the operator depends on the distance between the original plane and the analysis plane. The nonparaxial beam field in the plane of analysis after the axicon [34,35]: scalar model based on Rayleigh–Sommerfeld theory allows to obtain correct results at sufficiently close distances from the aperture of the optical element2π at which the diffraction occurs. Using the integral Z∞ Z ( ) Rayleigh–Sommerfeld transform of thezAP first type, we findexp the amplitude-phaseikL 1 distribution of the beam u(ρ, θ, zAP) = u0(r)τ(r) − ik + rdrdϕ (4) field in the plane of analysis after the 2axiconπ [34,35]: L L 0 0 ∞π z 2 exp()−ikL 1 ()ρ θ=AP ′() τ()  + ϕ 2 2 2 2 where u is distributionuz of the,, complexAP field urr amplitude before ikrdrd the axicon, L = ρ + r + z (4) 0 2π  LL AP − 2ρr cos(ϕ θ) denotes oblique length equal00 to the distance between the point at which the initial −′ wheredistribution u of theis complexdistribution amplitude of ofthe the fieldcomplexu0 is specifiedfield andamplitude a point in thebefore analysis the plane axicon, after 2222= ρ ++ −ρϕ()−θ L theLrzr axicon. ToAP calculate2cos the axial denotes intensity oblique of the outputlength field,equal weto usethe thedistance following between expression the point for at: 2 = 2 + 2 L r zAP. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 9

u′ whichAppl. Sci. the2020 initial, 10, 7911 distribution of the complex amplitude of the field is specified and a point in4 the of 9 analysis plane after the axicon. To calculate the axial intensity of the output field, we use the following 222=+ expression for L : LrzAP . When calculating amplitude-phase field field distribu distributionstions using the integral Rayleigh–Sommerfeld transformtransform within within the the nonparaxial nonparaxial scalar scalar model, model, it itis isnecessary necessary to toconsider consider the the efficiency efficiency and and error error of numericalof numerical methods methods when when working working with rapidly with rapidly oscillating oscillating functions. functions. One of such One practical of such methods practical ismethods Levin’s ismethod Levin’s [36], method which [36 allows], which one allows to calculate one to calculateintegrals integralswith complex with complexintegrands integrands of strongly of oscillatingstrongly oscillating functions. functions. According According to Levin’s to Levin’s method, method, the one-dimensional the one-dimensional oscillating oscillating integral integral is transformedis transformed into into an an ordinary ordinary differential differential equati equationon which which can can then then be be solved solved by by the collocation method [37]. [37]. The parameters of the axicon axicon directly directly determine th thee distribution of the beam field field in the the analysis analysis plane after the axicon, but do notnot aaffectffect thethe generalgeneral naturenature ofof the the e effectffect under under study. study. The The presence presence of of a abinary binary mask mask in frontin front of theof axiconthe axicon leads leads to the to fact th thate fact the that longitudinal the longitudinal dependence dependence of axial intensityof axial intensityof the Bessel of the beam Bessel does beam not obey does Equation not obey(3). Equation The distribution (3). The distribution of the axial of intensity the axial of intensity the Bessel of zone the Besselis modified zone is in modified such a way in thatsuch thea way width that of the the width central of corethe central and concentric core and ringsconcentric remains rings unchanged remains unchangedin a certain interval,in a certain while interval, the length while of thethe Bessel length zone of the decreases. Bessel Withzone didecreases.fferent parameters With different of the parametersbinary mask of transmission the binary mask function, transmission it is possible function to achieve, it is possible a sharp to change achieve in a thesharp edge change of the in axial the edgeintensity of the distribution axial intensity of the distribution Bessel zone of (Figure the Bessel2a–d). zone (Figure 2a–d).

(a) (b)

Figure 2.2. AxialAxial intensity intensity distributions distributions of aof Bessel a Bessel beam be andam the and gradient the gradient from the longitudinalfrom the longitudinal coordinate = coordinatez for various zAP parameters for various ofparameters the mask beforeof the themask axicon: before (a the) without axicon: binary (a) without mask, binaryd = 0mm mask,, D d 0 ; AP → ∞ mm,(b) with D →∞ binary; (b) mask,with binaryd = 4 mask,mm, Dd = 4 mm,;(c) withD →∞ binary; (c) with mask, binaryd = mask,0, D =dD7==mm;0, ( 7d)mm; with (d binary) with → ∞ binarymask, dmask,= 4, DdD===4,7 mm. 7 mm.

The analysis of the axial intensity distributions of of the the Bessel zone was carried out using the gradient of the scalar scalar function, function, wh whichich shows shows the the direction direction and and rate rate of ofits its most most rapid rapid increase/decrease increase/decrease [38,39]. [38,39 In]. FigureIn Figure 2, orange2, orange plots plots represent represent the value the valueof the offunction the function gradient gradient for each forpoint each in the point analysis in the plane analysis after plane after the zAP axicon. Based on the data, it follows that there was an abrupt change in the speed the zAP axicon. Based on the data, it follows that there was an abrupt change in the speed and direction and direction of the gradient of the function. of the gradient of the function. The position of the beam focusing plane after the axicon can be calculated using Equation (2) The position of the beam focusing plane after the axicon can be calculated using Equation (2) for for the length of the Bessel zone z , in which the input beam waist is substituted for beam height. the length of the Bessel zone z , inB which the input beam waist is substituted for beam height. The The results of this calculation areB also compared in Figure2, where the dotted vertical lines show results of this calculation are also compared in Figure 2, where the dotted vertical lines show the the positions of the focusing plane according to the ray model. The calculations based on the wave Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 9 Appl. Sci. 2020, 10, 7911 5 of 9 positions of the focusing plane according to the ray model. The calculations based on the wave theory correlate well with the ray model. It is important to note that the length and position of the Bessel theory correlate well with the ray model. It is important to note that the length and position of zone directly depend on the parameters of the amplitude mask: the left boundary along the the Bessel zone directly depend on the parameters of the amplitude mask: the left boundary along propagation direction is determined by the size of the inner aperture of the mask, and the right the propagation direction is determined by the size of the inner aperture of the mask, and the right boundary is determined by the outer aperture of the mask. The effective length of the Bessel zone is boundary is determined by the outer aperture of the mask. The effective length of the Bessel zone is defined as the difference between the positions corresponding to the outer and inner boundaries of defined as the difference between the positions corresponding to the outer and inner boundaries of the the amplitude mask. amplitude mask. 3. Experimental Implementation of the Bessel-Gaussian Beam 3. Experimental Implementation of the Bessel-Gaussian Beam

3.1.3.1. Experimental Experimental Setup Setup TheThe possibility possibility of of implementing implementing the the previo previouslyusly proposed proposed method method has has been been confirmed confirmed experimentally.experimentally. The source was a diode-pumped CW CW single-frequency laser laser ( (CoboltCobolt AB, AB, Solna, Solna, λ= ± Sweden)Sweden) with aa wavelengthwavelength of ofλ = 659.6659.60.3 0.3nm,nm, which which also also had had a spatial a spatial single-mode single-mode transverse transverse field 2 field structure ( TEM ) with a quality factor±2 M < 1.1. An afocal optical system was used to expand structure (TEM00) with00 a quality factor M < 1.1. An afocal optical system was used to expand the laser thebeam laser to abeam diameter to a diameter of 4 mm of (at 41 mm/e2 width).(at 1/e2 Then, width). a quasi-parallel Then, a quasi-parallel laser beam laser illuminated beam illuminated the mask theand mask the axicon. and the The axicon. axicon The AX2510-A axicon AX2510-A (Thorlabs, Newton,(Thorlabs, NJ, Newton, USA) had NJ, theUSA) following had the parameters: following α=  parameters:aperture diameter aperture 25.4 diameter mm, apex 25.4 angle mm,2 αapex0 = 160angle◦. The20 mask 160 (BMSTU,. The mask Moscow, (BMSTU, Russia) Moscow, was installed Russia) wasdirectly, installed close todirectly, the front close surface to the of front the axicon. surface The of maskthe axicon. we used The was mask glass we plane-parallel used was glass plates plane- with parallela circular plates aperture with 25.4 a circular mm in diameter,aperture 25.4 on one mm of in the diameter, surfaces ofon which one of a sutheffi cientlysurfaces thick of which layer ofa sufficientlychromium (thickτ < 0.01 layer) was of applied.chromium Figure ( τ<10.01 shows) was the applied. masks usedFigure in the1 shows research. the masks The parameters used in the of research.the mask The (clear parameters and inner apertures)of the mask were (clear designed and inner the sameapertures) as in modelingwere designed and mentioned the same inas thein modelingcaption of and Figure mentioned2. To register in the the caption axial intensity of Figure distribution 2. To register after the the axial axicon, intensity an afocal distribution optical system after x thewith axicon, 3 magnification an afocal optical was placed system in frontwith of3х amagnification sCMOS camera was CS2100M-USB placed in front (Thorlabs, of a sCMOS Newton, camera NJ, CS2100M-USBUSA) on the optical (Thorlabs, axis. Newton, NJ, USA) on the optical axis.

3.2.3.2. Results Results and and Analysis Analysis TheThe axial axial field ﬁeld intensity intensity distributions distributions were were acquir acquireded with with a step a step of 1 ofmm 1 mm along along the optical the optical axis. axis.The effectThe e ﬀofect a diaphragmed of a diaphragmed Gaussian Gaussian beam beam on the on axial the axial intensity intensity and and length length of the of theBessel Bessel zone zone is shown is shown in Figurein Figure 3a–d.3a–d. The The dots dots denote denote the the experimental experimental da data,ta, and and the the solid solid line denotes thethe approximatedapproximated experimentalexperimental data. data. It Itcan can be be seen seen that that the the expe experimentalrimental results results are are in in good good agreement agreement with with the the theoreticaltheoretical calculations calculations presented presented in in Section Section 2.2 .

(a) (b)

Figure 3. Cont. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 9

Appl. Sci. 2020, 10, 7911 6 of 9 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 9

Figure 3. Axial intensity distributions of a zero-order Bessel beam and gradient from the longitudinal

coordinate zAP for various parameters of the binary mask in front of the axicon: (a) without binary mask, d = 0 mm, D →∞; (b) with binary mask, d = 4 mm, D →∞; (c) with binary mask, dD==0, 7 mm; (d) with binary mask, dD==4, 7 mm.

z For the AP sections corresponding to the region of abrupt change in axial field intensity (vertical dashed lines in( cFigure) 3a–d), the cross-sections of field intensity are(d) shown in Figure 4. The resultsFigure of calculating 3. AxialAxial intensity intensity the gradient distributions distributions of the of of function a a zero-order zero-order are Be Besselpresentedssel beam beam inand and yellow gradient gradient lines from from in the Figure longitudinal longitudinal 3. The most interesting area from the entire dependence of the axial intensity distribution of the Bessel zone was coordinate zAPzAP for for various various parameters parameters of of the the binary bina maskry mask in front in front of the of axicon: the axicon: (a) without (a) without binary binary mask, the leadingmask,d = 0 mm, dand= 0D mm,trailing D;(→∞b edges.) with; (b) binary Whenwith binary mask, a Gaussian mask,d = 4 mm, dbeam= 4Dmm, hit anD;(→∞c axicon) with; (c) binarywith without binary mask, a mask,d = 0, thedDD=== 0,gradient7 mm; 7 of → ∞ → ∞ the functionmm;(d) with (d) with binaryhad binaryequal mask, mask,rised = and4, dDD== fall4,= 7 times, mm. 7 mm. Figure 3a. In the case when the outer or inner part of the Gaussian beam was diaphragmed, only one edge of the Bessel beam field intensity changed, as describedFor the inz APSectionz sections 2. With corresponding the external to thediaphr regionagm, of abruptthe leading change edge in axial in the field function intensity remained (vertical For the AP sections corresponding to the region of abrupt change in axial field intensity dashedunchanged, lines and in Figure with the3a–d), internal the cross-sections diaphragm, the of fieldtrailing intensity edge remain are showned unchanged. in Figure4. In The this results case, the of (vertical dashed lines in Figure 3a–d), the cross-sections of field intensity are shown in Figure 4. The calculatingrate of change the of gradient the intensity of the distribution function are profile presented in the in studied yellow linesareas in increased Figure3. by The more most than interesting 4 times. results of calculating the gradient of the function are presented in yellow lines in Figure 3. The most areaIn Figure from 3b the and entire Figure dependence 3c, one can ofthe see axial these intensity jumps in distribution the gradient of speed. the Bessel After zone installing was the the leading mask interesting area from the entire dependence of the axial intensity distribution of the Bessel zone was andwith trailing annular edges. aperture, When two a Gaussian sharp changes beam hitin anaxial axicon intensity without were a mask, obtained, the gradient Figure 3d. of theThe function rate of the leading and trailing edges. When a Gaussian beam hit an axicon without a mask, the gradient of hadchange equal also rise increased and fall by times, more Figure than3 4a. times. In the case when the outer or inner part of the Gaussian beam the function had equal rise and fall times, Figure 3a. In the case when the outer or inner part of the was diaphragmed,As mentioned onlyearlier, one Figure edge of 4 theshows Bessel the beamcross fieldsections intensity of the changed, field intensity as described distribution in Section of the2. Gaussian beam was diaphragmed, only one edge of the Bessel beam field intensity changed, as WithBessel the beam. external As a diaphragm,reference distribution, the leading we edge took in the cross function section remained that corresponded unchanged, andto Figure with the4a, described in Section 2. With the external diaphragm, the leading edge in the function remained internalwhen a Gaussian diaphragm, beam the with trailing a diameter edge remained of 4 mm without unchanged. apodization In this fell case, on thethe rateaxicon. of changeIt can be of seen the unchanged, and with the internal diaphragm, the trailing edge remained unchanged. In this case, the intensitythat there distribution were features profile of the in thezero-order studied Bessel areas increased function byin each more distribution than 4 times. that In Figureare of 3interest.b,c, one canThe rate of change of the intensity distribution profile in the studied areas increased by more than 4 times. seeabsolute these value jumps of in field the intensity gradient in speed. the analysis After installing plane for the the maskcases within Figure annular 4b–e aperture, corresponds two to sharp the In Figure 3b and Figure 3c, one can see these jumps in the gradient speed. After installing the mask changesintensity inprofile axial shown intensity in wereFigure obtained, 3a. Apodization Figure3d. of The the ratebeam of incident change alsoon the increased axicon did by morenot in than any with annular aperture, two sharp changes in axial intensity were obtained, Figure 3d. The rate of 4way times. affect the size or shape of the central maximum (the nucleus of the Bessel zone). change also increased by more than 4 times. As mentioned earlier, Figure 4 shows the cross sections of the field intensity distribution of the Bessel beam. As a reference distribution, we took the cross section that corresponded to Figure 4a, when a Gaussian beam with a diameter of 4 mm without apodization fell on the axicon. It can be seen that there were features of the zero-order Bessel function in each distribution that are of interest. The absolute value of field intensity in the analysis plane for the cases in Figure 4b–e corresponds to the intensity profile shown in Figure 3a. Apodization of the beam incident on the axicon did not in any way affect the size or shape of the central maximum (the nucleus of the Bessel zone).

(a) (b)

Figure 4. Cont.

(a) (b) Appl. Sci. 2020, 10, 7911 7 of 9 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 9

(e)

Figure 4. Cross sections of fieldfield intensity: (a) without binary mask,mask, dd==00mm,mm, DD →∞;;( (b) with with binary = →∞ == → ∞ mask, dd= 44mm,mm, D , Z,Zap ap= 240= 240 mm; mm; (c) with (c) with binary binary mask, mask, dD0,d = 0, 7 Dmm,= Z7 apmm, = 360 Z apmm;= 360 (d) with mm; ==→ ∞ == (binaryd) with mask, binary dD mask4, , d 7=mm,4, DZap= = 7240mm, mm; Zap (e)= with240 binary mm; (e mask,) with binarydD4, mask, 7 mm,d Z=ap4, = 360D = mm.7 mm, Zap = 360 mm. The change in relative position of the Bessel zone core in Figures 4 was associated with camera positioningAs mentioned accuracy earlier, during Figure the registration4 shows the of cross the sectionscross section of the of field the intensity field intensity distribution distribution. of the BesselHowever, beam. this As fact a referencedid not affect distribution, the analysis we and took interpretation the cross section of the that results. corresponded The same to can Figure be said4a, whenabout athe Gaussian triple symmetry beam with of a the diameter first maximum, of 4 mm withoutthe origin apodization of which fellhas onnothing the axicon. to do Itwith can the be seenamplitude that there diaphragm were features set in front of the of zero-order the axicon, Bessel and can function be explained in each distributionby the presence that areof aberrations of interest. Thein the absolute afocal optical valueof system field intensitylocated after in the the analysis axicon.plane for the cases in Figure4b–e corresponds to the intensity profile shown in Figure3a. Apodization of the beam incident on the axicon did not in any way4. Conclusions affect the size or shape of the central maximum (the nucleus of the Bessel zone). The change in relative position of the Bessel zone core in Figure4 was associated with camera The diffraction of a Gaussian beam on a binary mask and its propagation through an axicon to positioning accuracy during the registration of the cross section of the field intensity distribution. form a quasi-uniform axial intensity distribution of a zero-order Bessel beam were studied. With an However, this fact did not affect the analysis and interpretation of the results. The same can be said external or internal diaphragm of a Gaussian beam, only one edge of the axial intensity distribution about the triple symmetry of the first maximum, the origin of which has nothing to do with the of the Bessel beam field changed. The use of the mask with annular aperture allowed to obtain two amplitude diaphragm set in front of the axicon, and can be explained by the presence of aberrations in sharp edges in the distribution of the axial field intensity. In addition, the internal diaphragm of the the afocal optical system located after the axicon. input Gaussian beam allowed to exclude the influence of axicon apex angle quality on the formed 4.Bessel Conclusions beam. Thus, the choice of the parameters of the mask and the axicon allows one to control the length of the Bessel zone of the formed zero-order Bessel beam with a sharp drop in one or both edges The diffraction of a Gaussian beam on a binary mask and its propagation through an axicon to of the axial field intensity. This method and the laser optical system that implements it use a form a quasi-uniform axial intensity distribution of a zero-order Bessel beam were studied. With an commercially available element base and do not have significant restrictions on radiation resistance external or internal diaphragm of a Gaussian beam, only one edge of the axial intensity distribution when working with high-power radiation. The obtained research results will find application in of the Bessel beam field changed. The use of the mask with annular aperture allowed to obtain two various technologies of photonics and laser technology, such as optical tweezers, visualization of objects, material processing, etc. Appl. Sci. 2020, 10, 7911 8 of 9 sharp edges in the distribution of the axial field intensity. In addition, the internal diaphragm of the input Gaussian beam allowed to exclude the influence of axicon apex angle quality on the formed Bessel beam. Thus, the choice of the parameters of the mask and the axicon allows one to control the length of the Bessel zone of the formed zero-order Bessel beam with a sharp drop in one or both edges of the axial field intensity. This method and the laser optical system that implements it use a commercially available element base and do not have significant restrictions on radiation resistance when working with high-power radiation. The obtained research results will find application in various technologies of photonics and laser technology, such as optical tweezers, visualization of objects, material processing, etc.

Author Contributions: Conceptualization, N.S. and P.N.; methodology, M.K. and S.K.; software, M.G.; validation, N.S. and G.K.; formal analysis, P.N. and S.K.; investigation, N.S. and P.N.; resources, P.N. and M.K.; data curation, M.G.; writing—original draft preparation, N.S., P.N. and G.K.; writing—review and editing, G.K. and M.K.; visualization, N.S.; supervision, S.K.; project administration, N.S. and P.N.; funding acquisition, P.N. and M.K. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Russian Foundation for Basic Research, grant number 18-38-20155, and Ministry of Science and Higher Education of the Russian Federation, grant number 0705-2020-0041. Conﬂicts of Interest: The authors declare no conﬂict of interests.

References

1. Saraeva, I.N.; Kudryashov, S.I.; Rudenko, A.A.; Zhilnikova, M.I.; Ivanov, D.S.; Zayarny, D.A.; Simakin, A.V.; Ionin, A.A.; Garcia, M.E. Effect of fs/ps laser pulsewidth on ablation of metals and silicon in air and liquids on their nanoparticle yields. Appl. Surf. Sci. 2019, 470, 1018–1034. [CrossRef] 2. Galaktionov, I.; Kudryashov, A.; Sheldakova, J.; Nikitina, A. Laser beam focusing through the scattering medium by means of adaptive optics. In SPIE Proceedings Vol. 10073: Adaptive Optics and Wavefront Control for Biological Systems III, Proceedings of the SPIE BiOS, San Francisco, CA, USA, 28–30 January 2017; Bifano, T.G., Kubby, J., Gigan, S., Eds.; SPIE—International Society for Optics and Photonics: Bellingham, WA, USA, 2017. [CrossRef] 3. Saj, W.M. Light focusing on a stack of metal-insulatormetal waveguides sharp edge. Opt. Express 2009, 17, 13615–13623. [CrossRef][PubMed] 4. Stsepuro, N.G.; Krasin, G.K.; Kovalev, M.S.; Pestereva, V.N. Determination of the Point Spread Function of a Computer-Generated Lens Formed by a Phase Light Modulator. Opt. Spectrosc. 2020, 128, 1036–1040. [CrossRef] 5. Simonov, A.N.; Rombach, M.C. Sharp-focus image restoration from defocused images. Opt. Lett. 2009, 34, 2111–2113. [CrossRef][PubMed] 6. Dorn, R.; Quabis, S.; Leuchs, G. Sharper Focus for a Radially Polarized Light Beam. Phys. Rev. Lett. 2003, 91, 233901. [CrossRef][PubMed] 7. Nosov, P.A.; Piskunov, D.E.; Shirankov, A.F. Combined laser variosystems paraxial design for longitudinal movement of a Gaussian beam waist. Opt. Express 2020, 28, 5105–5118. [CrossRef] 8. Galkin, M.L.; Nosov, P.A.; Kovalev, M.S.; Verenikina, N.M. Calculation and analysis of the laser beam field distribution formed by a real optical system. J. Phys. Conf. Ser. 2018, 1096, 012120. [CrossRef] 9. Miks, A.; Pokorny, P. Fundamental design parameters of two-component optical systems: Theoretical analysis. Appl. Opt. 2020, 59, 1998–2003. [CrossRef] 10. Mcleod, J.H. The Axicon: A New Type of Optical Element. J. Opt. Soc. Am. 1954, 44, 592–597. [CrossRef] 11. Yew, E.Y.S.; Sheppard, C.J.R. Tight focusing of radially polarized Gaussian and Bessel-Gauss beams. Opt. Lett. 2007, 32, 3417–3419. [CrossRef] 12. Kotlyar, V.V.; Stafeev, S.S.; Porfirev, A.P. Tight focusing of an asymmetric Bessel beam. Opt. Commun. 2015, 357, 45–51. [CrossRef] 13. Summers, A.M.; Yu, X.; Wang, X.; Raoul, M.; Nelson, J.; Todd, D.; Zigo, S.; Lei, S.; Trallero-Herrero, C.A. Spatial characterization of Bessel-like beams for strong-field physics. Opt. Express 2017, 25, 1646–1655. [CrossRef] 14. Zhang, Z.; Zeng, X.; Miao, Y.; Fan, Y.; Gao, X. Focusing properties of vector Bessel–Gauss beam with multiple-annular phase wavefront. Optik 2018, 157, 240–247. [CrossRef] Appl. Sci. 2020, 10, 7911 9 of 9

15. Dudutis, J.; Geˇcys,P.; Raˇciukaitis,G. Non-ideal axicon-generated Bessel beam application for intra-volume glass modification. Opt. Express 2016, 24, 28433–28443. [CrossRef][PubMed] 16. Mcgloin, D.; Dholakia, K. Bessel beams: Diffraction in a new light. Contemp. Phys. 2005, 46, 15–28. [CrossRef] 17. Herman, R.M.; Wiggins, T.A. Production and uses of diffractionless beams. J. Opt. Soc. Am. A 1991, 8, 932–942. [CrossRef] 18. Gorelick, S.; Paganin, D.; Korneev, D.; de Marco, A. Hybrid refractive-diffractive axicons for Bessel-beam multiplexing and resolution improvement. Opt. Express 2020, 28, 12174–12188. [CrossRef] 19. Kharitonov, S.I.; Khonina, S.N.; Volotovskiy, S.G.; Kazanskiy, N.L. Caustics of the vortex beams generated by vortex lenses and vortex axicons. J. Opt. Soc. Am. A 2020, 37, 476–482. [CrossRef] 20. Fahrbach, F.O.; Rohrbach, A. A line scanned light-sheet microscope with phase shaped self-reconstructing beams. Opt. Express 2010, 18, 24229–24244. [CrossRef] 21. Chang, B.-J.; Dean, K.M.; Fiolka, R. A systematic and quantitative comparison of lattice and Gaussian light-sheets. Opt. Express 2020, 28, 27052–27077. [CrossRef] 22. Chang, B.-J.; Kittisopikul, M.; Dean, K.M.; Roudot, P.; Welf, E.S.; Fiolka, R. Universal light-sheet generation with field synthesis. Nat. Methods 2019, 16, 235–238. [CrossRef][PubMed] 23. Meinert, T.; Rohrbach, A. Light-sheet microscopy with length-adaptive bessel beams. Biomed. Opt. Express 2019, 10, 670–681. [CrossRef][PubMed] 24. Xiong, B.; Han, X.; Wu, J.; Xie, H.; Dai, Q. Improving axial resolution of Bessel beam light-sheet fluorescence microscopy by photobleaching imprinting. Opt. Express 2020, 28, 9464–9476. [CrossRef] 25. Yu, X.; Zhang, M.; Lei, S. Multiphoton polymerization using femtosecond bessel beam for layerless three-dimensional printing. J. Micro Nanomanufacturing 2017, 6, 010901. [CrossRef] 26. Stoian, R.; Bhuyan, M.K.; Rudenko, A.; Colombier, J.-P.; Cheng, G. High-resolution material structuring using ultrafast laser non-diffractive beams. Adv. Phys. X 2019, 4, 1659180. [CrossRef] 27. Chong, A.; Renninger, W.H.; Christodoulides, D.N.; Wise, F.W. Airy-Bessel wave packets as versatile linear light bullets. Nat. Photonics 2010, 4, 103–106. [CrossRef] 28. Szulzycki, K.; Savaryn, V.; Grulkowski, I. Generation of dynamic Bessel beams and dynamic bottle beams using acousto-optic effect. Opt. Express 2016, 24, 23977–23991. [CrossRef] 29. Monsoriu, J.A.; Furlan, W.D.; Andrés, P.; Lancis, J. Fractal conical lenses. Opt. Express 2006, 14, 9077–9082. [CrossRef] 30. Herrmann, J.; Wilhelmi, B. Lasers for Ultrashort Light Pulses; North-Holland: Amsterdam, The Netherlands, 1987. 31. Ustinov, A.V.; Khonina, S.N. Calculating the Complex Transmission Function of Refractive Axicons. Opt. Mem. Neural Netw. 2012, 21, 133–144. [CrossRef] 32. Arlt, J.; Dholakia, K.; Soneson, J.; Wright, E.M. Optical dipole traps and atomic waveguides based on Bessel light beams. Phys. Rev. A 2001, 63, 063602. [CrossRef] 33. Brzobohatý, O.; Cižmˇ ár, T.; Zemánek, P. High quality quasi-Bessel beam generated by round-tip axicon. Opt. Express 2008, 16, 12688–12700. [CrossRef][PubMed] 34. Khonina, S.N.; Ustinov, A.V.; Kovalev, A.A.; Volotovsky, S.G. Propagation of the radially-limited vortical beam in a near zone. Part I. Calculation algorithms. Comput. Opt. 2010, 34, 315–329. 35. Khonina, S.N.; Ustinov, A.V.; Kovalev, A.A.; Volotovsky, S.G. Propagation of the radially-limited vortical beam in a near zone. Part II. Results of simulation. Comput. Opt. 2010, 34, 330–339. 36. Levin, D. Procedures for computing one and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 1982, 38, 531–538. [CrossRef] 37. Itô, K. Encyclopedic Dictionary of Mathematics; MIT Press: Cambridge, MA, USA, 1980; p. 1139. 38. Kaplan, W. The Gradient Field; Addison-Wesley: Reading, MA, USA, 1991; pp. 183–185. 39. Morse, P.M.; Feshbach, H. The Gradient. In Methods of Theoretical Physics, Part I; McGraw-Hill: New York, NY, USA, 1953; pp. 31–32.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).